Properties

Label 768.3.b.c
Level $768$
Weight $3$
Character orbit 768.b
Analytic conductor $20.926$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,3,Mod(127,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{3} q^{5} + 2 \beta_{2} q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{3} q^{5} + 2 \beta_{2} q^{7} + 3 q^{9} + 4 \beta_1 q^{11} + \beta_{3} q^{13} - \beta_{2} q^{15} + 10 q^{17} - 12 \beta_1 q^{19} - 6 \beta_{3} q^{21} + 8 \beta_{2} q^{23} + 21 q^{25} - 3 \beta_1 q^{27} - 13 \beta_{3} q^{29} + 2 \beta_{2} q^{31} - 12 q^{33} - 8 \beta_1 q^{35} - 13 \beta_{3} q^{37} - \beta_{2} q^{39} - 58 q^{41} - 28 \beta_1 q^{43} + 3 \beta_{3} q^{45} + 20 \beta_{2} q^{47} + q^{49} - 10 \beta_1 q^{51} + 37 \beta_{3} q^{53} + 4 \beta_{2} q^{55} + 36 q^{57} - 52 \beta_1 q^{59} + 13 \beta_{3} q^{61} + 6 \beta_{2} q^{63} - 4 q^{65} - 4 \beta_1 q^{67} - 24 \beta_{3} q^{69} + 46 q^{73} - 21 \beta_1 q^{75} + 24 \beta_{3} q^{77} + 34 \beta_{2} q^{79} + 9 q^{81} - 28 \beta_1 q^{83} + 10 \beta_{3} q^{85} + 13 \beta_{2} q^{87} - 82 q^{89} - 8 \beta_1 q^{91} - 6 \beta_{3} q^{93} - 12 \beta_{2} q^{95} + 2 q^{97} + 12 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} + 40 q^{17} + 84 q^{25} - 48 q^{33} - 232 q^{41} + 4 q^{49} + 144 q^{57} - 16 q^{65} + 184 q^{73} + 36 q^{81} - 328 q^{89} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{12}^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0 −1.73205 0 2.00000i 0 6.92820i 0 3.00000 0
127.2 0 −1.73205 0 2.00000i 0 6.92820i 0 3.00000 0
127.3 0 1.73205 0 2.00000i 0 6.92820i 0 3.00000 0
127.4 0 1.73205 0 2.00000i 0 6.92820i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.3.b.c 4
3.b odd 2 1 2304.3.b.l 4
4.b odd 2 1 inner 768.3.b.c 4
8.b even 2 1 inner 768.3.b.c 4
8.d odd 2 1 inner 768.3.b.c 4
12.b even 2 1 2304.3.b.l 4
16.e even 4 1 12.3.d.a 2
16.e even 4 1 192.3.g.b 2
16.f odd 4 1 12.3.d.a 2
16.f odd 4 1 192.3.g.b 2
24.f even 2 1 2304.3.b.l 4
24.h odd 2 1 2304.3.b.l 4
48.i odd 4 1 36.3.d.c 2
48.i odd 4 1 576.3.g.e 2
48.k even 4 1 36.3.d.c 2
48.k even 4 1 576.3.g.e 2
80.i odd 4 1 300.3.f.a 4
80.j even 4 1 300.3.f.a 4
80.k odd 4 1 300.3.c.b 2
80.q even 4 1 300.3.c.b 2
80.s even 4 1 300.3.f.a 4
80.t odd 4 1 300.3.f.a 4
112.j even 4 1 588.3.g.b 2
112.l odd 4 1 588.3.g.b 2
144.u even 12 1 324.3.f.a 2
144.u even 12 1 324.3.f.g 2
144.v odd 12 1 324.3.f.d 2
144.v odd 12 1 324.3.f.j 2
144.w odd 12 1 324.3.f.a 2
144.w odd 12 1 324.3.f.g 2
144.x even 12 1 324.3.f.d 2
144.x even 12 1 324.3.f.j 2
240.t even 4 1 900.3.c.e 2
240.z odd 4 1 900.3.f.c 4
240.bb even 4 1 900.3.f.c 4
240.bd odd 4 1 900.3.f.c 4
240.bf even 4 1 900.3.f.c 4
240.bm odd 4 1 900.3.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 16.e even 4 1
12.3.d.a 2 16.f odd 4 1
36.3.d.c 2 48.i odd 4 1
36.3.d.c 2 48.k even 4 1
192.3.g.b 2 16.e even 4 1
192.3.g.b 2 16.f odd 4 1
300.3.c.b 2 80.k odd 4 1
300.3.c.b 2 80.q even 4 1
300.3.f.a 4 80.i odd 4 1
300.3.f.a 4 80.j even 4 1
300.3.f.a 4 80.s even 4 1
300.3.f.a 4 80.t odd 4 1
324.3.f.a 2 144.u even 12 1
324.3.f.a 2 144.w odd 12 1
324.3.f.d 2 144.v odd 12 1
324.3.f.d 2 144.x even 12 1
324.3.f.g 2 144.u even 12 1
324.3.f.g 2 144.w odd 12 1
324.3.f.j 2 144.v odd 12 1
324.3.f.j 2 144.x even 12 1
576.3.g.e 2 48.i odd 4 1
576.3.g.e 2 48.k even 4 1
588.3.g.b 2 112.j even 4 1
588.3.g.b 2 112.l odd 4 1
768.3.b.c 4 1.a even 1 1 trivial
768.3.b.c 4 4.b odd 2 1 inner
768.3.b.c 4 8.b even 2 1 inner
768.3.b.c 4 8.d odd 2 1 inner
900.3.c.e 2 240.t even 4 1
900.3.c.e 2 240.bm odd 4 1
900.3.f.c 4 240.z odd 4 1
900.3.f.c 4 240.bb even 4 1
900.3.f.c 4 240.bd odd 4 1
900.3.f.c 4 240.bf even 4 1
2304.3.b.l 4 3.b odd 2 1
2304.3.b.l 4 12.b even 2 1
2304.3.b.l 4 24.f even 2 1
2304.3.b.l 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T - 10)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 768)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$41$ \( (T + 58)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2352)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4800)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 5476)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8112)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T - 46)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 13872)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2352)^{2} \) Copy content Toggle raw display
$89$ \( (T + 82)^{4} \) Copy content Toggle raw display
$97$ \( (T - 2)^{4} \) Copy content Toggle raw display
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